Transport Methods for Nuclear Reactor Analysis

1. Transport Methods for Nuclear Reactor Analysis Marvin L. AdamsTexas A&M University mladams@tamu.edu Computational Methods in Transport Tahoe City, September 11-16, 2004

2. Acknowledgments Thanks to Frank for organizing this! Kord Smith taught me much of what I know about modern nuclear reactor analysis.

3. 3 Outline Bottom Line Problem characteristics and solution requirements Modern methodology Results: Amazing computational efficiency! Summary

4. 4 Modern methods are dramatically successful for LWR transport problems. Today�s codes calculate power density (W/cm3) in each of the 30,000,000 fuel pellets critical rod configuration or boron concentration nuclide production and depletion as a function of time for a full one- to two-year cycle including off-normal conditions including coupled heat transfer and coolant flow with accuracy of a few % using a $1000 PC in < 4 hours This is phenomenal computational efficiency!

5. 5 Outline Bottom Line Problem characteristics and solution requirements Geometry is challenging Physics is challenging Requirements are challenging Modern methodology Results: Amazing computational efficiency! Summary

6. 6 Reactor geometry presents challenges. Fuel pins are simple (cylindrical tubes containing a stack of pellets) but there are 50,000 of them and we must compute power distribution in each one! Structural materials are complicated grid spacers, core barrel, bundle cans Instrumentation occupies small volumes

7. 7 For physics, it helps to remember a simplified neutron life cycle.

8. 8 It also helps to know something about the answer. Cartoon (not actual result) of basic dependence on energy in thermal reactor Fission-spectrum-ish at high energies 1/E-ish in intermediate energies Maxwellian-ish at low energies 10 orders of magnitude in domain and range

9. 9 Neutron-nucleus interaction physics presents challenges: cross sections are wild! Resonances: s changes by 2-4 orders of magnitude with miniscule changes in neutron energy (really total kinetic energy in COM frame) arise from discrete energy levels in compound nucleus effectively, become shorter and broader with increasing material temperature (because of averaging over range of COM kinetic energies) Bottom line: s�s depend very sensitively on neutron energy and material temperature!

10. 10 A milder challenge: scattering is anisotropic. Scattering is isotropic in the center-of-mass frame for: light nuclides neutron energies below 10s of keV Not for higher energies. Not for heavier nuclides. Almost never isotropic in lab frame!

11. 11 Temperature dependence makes this a coupled-physics problem.

12. 12 Depletion and creation of nuclides adds to the challenge. Example: depletion of �burnable absorber� (such as Gd) Some fuel pellets start with Gd uniformly distributed Very strong absorber of thermal n�s Thermal n�s enter from coolant ? n-Gd absorptions occur first in outer part of pellet Gd depletion eats its way inward over time Example: U238 depletion and Pu239 buildup Similar story Most n capture in U238 is at resonance energies, where S is huge At resonance energies, most n�s enter fuel from coolant ? captures occur first in outer part of pellet U239 ? Np239 ? Pu239, and Pu239 is fissile �Rim effect�

13. 13 Transient calculations present further challenges. Delayed neutrons are important! Small fraction of n�s from fission are released with significant time delays prompt neutrons (>99%) are released at fission time a released neutron takes < 0.001 s to either leak or be absorbed delayed neutrons (<1%) are released 0.01 � 100 s after fission they are emitted during decay of daughters of fission products (delayed-neutron precursors) Doesn�t affect steady state. Delayed neutrons usually dominate transient behavior. slightly supercritical reactor would be subcritical without dn�s subcritical reactor behavior limited by decay of slowest �precursor� must calculate precursor concentrations and decay rates as well as neutron flux (and heat transfer and fluid flow)

14. 14 Solution requirements are challenging. To license a core for a cycle (1-2 years), must perform thousands of full-core calculations dozens of depletion steps hundreds of configurations per step Each calculation must provide enormous detail axial distribution of power for each of 50,000 pins depletion and production in hundreds of regions per pin includes heat transfer and coolant flow includes search for critical (boron concentration or rod position) Transient calculations are also required Simulators require incredible computational efficiency (real-time simulation of entire plant)

15. 15 Outline Bottom Line Problem characteristics and solution requirements Modern methodology Divide & Conquer Sophisticated averaging Factorization / Superposition Coupling, searches, and iterations Results: Amazing computational efficiency! Summary

16. 16 Divide-and-Conquer approach relies on multiple levels of calculation.

17. 17 How can 2-group diffusion give good answers to such complicated transport problems? Homogenization Theory: Low-order model can reproduce (limited features of) any reference high-order solution. Consider a reference solution generated by many-group fine-mesh transport for heterogeneous region. 2-group coarse-mesh diffusion on a homogenized region can reproduce: reaction rates in coarse cell net flow across each surface of coarse cell �Discontinuity Factors� make this possible! 2-group diffusion parameters come from fairly accurate reference solution: f(r,E) from single-assembly calculation Diffusion is reasonably accurate given large homogeneous regions.

18. 18 Assembly-level calculation has very high fidelity. 2D long-characteristics transport Scattering and fission sources assumed constant (�flat�) in each mesh region Essentially exact geometry Dozens of energy groups Thousands of flat-source mesh regions

19. 19 Fine-mesh fine-group assembly-level solution is used to average the S�s. S�s are averaged over �fast� and �thermal� energy ranges: thermal: (0,1) eV fast (1,10000000) eV Assemblies are �homogenized� by spatially averaging their S�s: If averaging function has same �shape� as the real solution, then averaged S�s produce the correct reaction rates in low-order calculation. Assuming that net flow rates are correct ...

20. 20 Even perfectly averaged S�s are not enough! Also need correct net leakages. Even with perfectly averaged S�s, the homogenized problem cannot produce correct reaction rates and correct leakages. The solution is to specify a discontinuity in the scalar flux at assembly surfaces, using a �discontinuity factor:� This is what makes homogenization work!

21. 21 We generate DF�s from the single-assembly problems. Single Assembly: uses reflecting boundary fine-mesh fine-group transport generates �exact� f this generates homogenized 2-group S�s then solve homogenized single-assembly problem with low-order operator (coarse-mesh 2-group diffusion) DF is ratio of exact to low-order solution on each surface Core Level: we know that exact heterogeneous solution is continuous in each coarse mesh, this is approximated as the low-order solution times the DF for that assembly and surface continuity of this quantity means discontinuity of low-order solution (unless neighboring assemblies have the same DF)

22. 22 Global calculation must produce pin-by-pin powers as well as coarse-mesh reaction rates. Pin power reconstruction is done using �form functions.� Basic idea: assume that depends weakly on assembly boundary conditions. We tabulate this �form function� for each fuel pin in the single-assembly calculation, then use it to generate pin powers after each full-core calculation.

23. 23 In the core, every assembly is different. Core-level code needs S�s and F�s as functions of: fuel temperature coolant temperature boron concentration void fraction burnup various history effects etc. Assembly-level code produces tables using �branch cases.� Basic idea: define base-case parameter values; run base case and tabulate change one parameter; re-run. Generates dS/dp for this p. repeat for all parameters

24. 24 Still must discretize 2-group diffusion accurately on coarse homogenized regions. Lots of ways to do this well enough. Typical modern method: high-order polynomials for fast flux (4th-order, e.g.) continuity conditions and spatial-moment equations determine the unknowns thermal equation is solved semi-analytically transverse integration produces coupled 1D equations each is solved analytically (giving sinh and cosh functions) transverse-leakage terms are approximated with quadratic polynomials Result is quite accurate for the large homogenized regions used in practice.

25. 25 Outline Bottom Line Problem characteristics and solution requirements Modern methodology Divide & Conquer Sophisticated averaging Factorization / Superposition Coupling, searches, and iterations Results: Amazing computational efficiency! Summary

26. 26 Coupling and search is rolled into eigenvalue iteration in practice. Guess k, fission source, temperatures, and boron concentration. Solve 2-group fixed-source problem new k, fission source, region-avg f�s, and surface leakages Use surface leakages and region-avg f�s to define CMFD equations. Use CMFD equations to iterate on k fission source temperatures (coupled to heat transfer and fluid flow) boron concentration Update high-order solution; repeat. This is incredibly fast!

27. 27 Results demonstrate truly amazing computational efficiency. Assembly-level code: 1600 2D transport calculations per PWR assembly hundreds of flat-source regions dozens of energy groups dozens to hundreds of directions per group; 0.2-mm ray spacing total run time < 1 hr (<2 s per 2D transport calculation) on cheap PC Core typically has 3-5 different kinds of assemblies. Core-level code: thousands of 3D diffusion calculations per cycle 200 x 25 coarse cells high-order polynomial / analytic function coupled to heat transfer and fluid flow; critical search done pin-power reconstruction < 4 s per 3D problem on cheap PC k errors <0.1%. Pin-power errors <5% (RMS avg < 1%)

28. 28 Summary Reactor analysis methods are quite mature for commercial LWRs. They are really, really fast! They work very well for all-uranium cores. Still some challenges for MOX cores.